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Question

# Determine whether the given series converges or diverges. $\sum_{n=1}^{\infty} \frac{1}{n^{2} \sqrt{n}}$

Solution

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Rewrite this $p$-series into exponential form.

\begin{align*} \displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^2\sqrt{n}} &= \displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^2\cdot n^{1/2}}\\ &= \displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^{5/2}} \end{align*}

Since $p=\dfrac{5}{2}>1$, then we conclude that this series converges.

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